Holographic Code Rate

TL;DR

This paper investigates holographic codes built on hyperbolic tessellations with perfect tensors, demonstrating how hyperbolic geometry constrains code rate and guarantees quantum error correction for most inflation rules.

Contribution

It establishes bounds on the holographic code rate using hyperbolic geometry and shows that most inflation rules on regular tessellations enable quantum error correction.

Findings

Hyperbolic geometry bounds the code rate.

Most inflation rules support quantum error correction.

The tile completion rule exceeds code rate one, but others do not.

Abstract

Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation rule protect quantum information stored in the bulk from errors on the boundary provided the code rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum error correction for codes grown with any inflation rule on all regular hyperbolic tessellations in a class whose size grows exponentially with the rank of the perfect tensors for rank five and higher. For the tile completion inflation rule, holographic triangle codes have code rate more than one but all others perform quantum error correction.